Chapter 7: Rotational Motion and the Law of Gravity

Section 1: Measuring Rotational Motion
Rotational motion occurs when an object spins about its axis of rotation.
When discussing rotational motion, the radius (distance from the center) is expressed as r, the angle of rotation is expressed as theta, and the arc length is expressed as s.
When dealing with rotational motion equations, the angles are measured in radians. Radians can be converted into degrees using
$\displaystyle \theta$ representing any angle
$\displaystyle \theta _{rad}=\frac{pi}{180} \theta _{deg}$
Radians can be expressed by the other terms using the equation
$\displaystyle \theta=\frac{s}{r}$
This equation is also useful for expressing angular displacement (the angle through which a point, line or body is rotated in a specified direction and about a specified axis). When used for this purpose, the equation is expressed as
$\displaystyle \Delta\theta=\frac{\Delta s}{r}$
where
$\displaystyle angular\ displacement (in\ radians)=\frac{change\ in\ arc\ length}{distance\ from\ axis}$
Usually, when a rotating object is viewed above, it is considered positive when the point rotates counterclockwise and negative when it rotates clockwise.

Section 2: Tangential and Centripetal Acceleration

Tangential Speed: Is the speed of any point rotating about an axis, also called the instantaneous linear speed of that point.
To find tangential speed, you need to take the distance from the axis multiplied by the angular speed of said object.
$\displaystyle V_t=r\omega$
Note: The omega is the instantaneous linear speed instead of the average angular speed because the time interval is very small.

Suppose there is an amusment park carousel and it begins to speed up. The various animals on it begin to experience an angular acceleration. The linear acceleration that is related to the angular acceleration these objects feel is tangent to the circular path that they are travelling in is called Tangential acceleration.
$\displaystyle a_t=r\alpha$
When using this equation, it must be expressed in radians to work and the angular acceleration is the instantaneous angular acceleration due to the very small time interval.

Centripetal Acceleration: Is an acceleration due to a change in direction, not due to a change in magnitude.
If a man is running in a circle at a constant speed, he is experiencing a centripetal acceleration because his direction is changing.
$\displaystyle a_c=\frac {v_t^2} {r}$
$\displaystyle a_c=r\omega^2$

Review
1. A discus thrower spins in a circle before releasing the discus with a tangential speed of 9.0 m/s. What is the angular speed of the spining discus thrower? Assume the discus is .75 from the thrower's axis of rotation.
2. What is a tire's angular acceleration if the tangential acceleration at a radius of .15 m is 9.4 x 10^-2?
3. A girl sits on a tire swing. Her father pushes her so that her centripetal acceleration of .512 rad/s. If the length of the rope is 2.1 m long what is the girls tangential speed?

Answers
1. 12 rad/s
2. .63 rad/s^2
3. 2.5 m/s

3: Causes of Circular Motion
Newton's second Law for rotation
In Section 2, we understood that there exists a relationship between the net torque of an object and the angular acceleeration to the object. While Newton's second law, which explains that an object's net force relates with the object's translational acceleration.
Newton's Second Law for Rotational Motion
$\displaystyle T_net=I\alpha$
net torque=moment of inertia X angular acceleration
This equation is simliar to the transilational motion
$\displaystyle F=ma$

Anfular Momentum, which is the rotational momentum associated when an object while rotating due to the fact it possees inertia.
$\displaystyle L=I\omega$
The unit of angular momentum is
$\displaystye kgm^2/s$

Rotatinonal Kinetic Energy is the kinetic energy associated with an object's angular speed.
$\displaystyle KE_rot=\frac{1}{2}I\omega^2$
Mechanical energy however may be conserved, howere we then uses the following equations:
$\displaystyle ME=KE_trans+KE_rot+PE_g$
$\displaystyle ME=\frac{1}{2}mv^2+\frac{1}{2}I\omega^2+mgh$

4. Simple Machines
The six baisc simple machines that modfies force are lever, pulley, inclined plane, wheel and axel, wedge, and screw.

Mechanincal Advantage is the ratio how large the output force is related to the input force
$\displaystyle MA=\frac{F_out}{F_in}$
$\displaystyle MA=\frac{F_out}{F_in}=\frac{d_in}{d_out}$

Efficiency of a machine is a measure of how much input energy is lost compared with how much energy is used to perfom work on an object.
$\displaystyle eff=\frac{W_out}{W_in}$

If a machine is frictionless, then the mechanical energy is conserved. If the mechanical efficiency is 1, then it is 100 percent. Since all machines have friction, the efficiency of real machines will always be less than 1.